Hill’s equations describe the linearized relative motion of deputy spacecraft with respect to the chief spacecraft which moves around the Earth in a circular orbit. The reachable set is defined as the set of all possible final positions of the deputy spacecraft that can be reached by a control force under certain constraints. The computation of the reachable set has been a focus of recent study, and different numerical methods are proposed. In this article, we use rigorous mathematical methods to study the structure and properties of the reachable set under the energy-limited constraint. We give an analytical expression for the computation of reachable set by using the methods of Hilbert space theory. According to the obtained analytical expression, the shape, volume, surface area and other properties of the reachable set are studied. In the end, the conclusions are verified numerically.

希尔的方程式描述了副航天器相对于绕圆形轨道绕地球运动的主要航天器的线性相对运动。可到达集合被定义为控制力在一定约束下可以到达的副航天器所有可能的最终位置的集合。可达集的计算一直是最近研究的重点，并提出了不同的数值方法。在本文中，我们使用严格的数学方法来研究能量受限约束下可到达集合的结构和性质。我们使用希尔伯特空间理论的方法给出了可达集计算的解析表达式。根据获得的解析表达式，研究了可达集合的形状，体积，表面积和其他属性。到底，